Illinois background check resources usually start with criminal histories. To get an actual criminal record file about another person, you can put in a name-only request (as opposed to a fingerprint application) to the State Police Department. The fee for each search is $16 and you can get the forms from their website, though you cannot actually do a records search online. On a related note, you can also get copies of court records from the Administrative Office of Illinois Courts, and you can also look up current inmates with the Illinois Department of Corrections.
Other Family Records
You can't get marriage or divorce records directly, but the Vital Records office can issue "verifications" about a record that will just verify that the record exists. If you need a copy of the actual document, you have to contact the county clerk office where it was registered.